Given a scalar function $s:S^2\to \mathbb{R}$, and the induced Euclidean metric $g$ on $S^2$, when can the sphere, equipped with metric $e^sg$, be isometrically immersed in $\mathbb{R}^3$? Is there a characterization in terms of $s$?
I believe that for the analogous problem in 2D, the disk can be isometrically immersed in the plane if the conformal factor is harmonic.